Abstract not found

In this paper, quantile regression methods are suggested for a class of smooth coefficient time series models. We employ a local linear fitting scheme to estimate the smooth coefficients in the quantile framework. The programming involved in the local linear quantile estimation is relatively simple and it can be modified with few efforts from the existing programs for the linear quantile model. We derive the local Bahadur representation of the local linear estimator for α-mixing time series and establish the asymptotic normality of the resulting estimator. Also, a bandwidth selector based on the nonparametric version of the Akaike information criterion is proposed, together with a consistent estimate of the asymptotic covariance matrix. The asymptotic behaviors of the estimator at the boundaries are examined. A comparison of the local linear quantile estimator with the local constant estimator is presented. A simulation study is carried out to illustrate the performance of the estimates. An empirical application of the model to the exchange rate time series data and the well-known Boston house price data further demonstrates the potential of the proposed modeling procedures. KEY WORDS: Bandwidth selection; boundary effect; covariance estimation; kernel smoothing methods; nonlinear time series; quantile regression; value-at-risk; varying coefficients.

i . As a particular case, we consider the case ae(x) = jxj p . In this case, we show that if E[kZk p +kZk 2 ] ! 1; either p ? 1=2 or m 2; and some other regularity conditions hold, then n 1=2 ( ` n \Gamma ` 0 ) converges in distribution to a normal limit. For m = 1 and p = 1=2, n 1=2 (log n) \Gamma1=2 ( ` n \Gamma ` 0 ) converges in distribution to a normal limit. For m = 1 and 1=2 ? p ? 0, n<F7.92

.<F3.82e+05> In linear regression, an important role is played by the least quantile of squares (LQS) estimate, which involves the minimization of the<F2.934e+05><F3.82e+05> qth smallest squared residual for a given set of data. This function is nondi#erentiable and nonconvex and may have a large number of local minima. This paper is mainly concerned with the e#cient calculation of the global solution, and some di#erent approaches are considered.<F4.005e+05> Key words.<F3.82e+05> linear regression, LQS estimate, Chebyshev approximation<F4.005e+05> AMS subject classifications.<F3.82e+05> 62J05, 65D10<F4.005e+05> PII.<F3.82e+05> S1064827595283768<F4.61e+05> 1. Introduction.<F4.492e+05> The problem of fitting a linear model to data usually involves the solution of an overdetermined system of linear equations which can be expressed as<F3.774e+05><F4.492e+05> (1.1)<F3.774e+05><F4.61e+05> Ax<F4.634e+05> #<F4.61e+05><F3.774e+05> b,<F4.492e+05> where<F4.61e+05> x<F4.634e+05> #<F4.492e+05>...

A test for globally testing the four assumptions of the linear model is proposed. The test can be viewed as a Neyman’s smooth test and it only relies on the residual vector. The components of the global test statistic could be utilized to gain insights into which assumptions have been violated if the global procedure indicates that there is a breakdown in at least one of the four assumptions. The procedure could be used in conjunction with the usual graphical methods, and it is simple enough to be implemented by beginning statistics students. The procedure is demonstrated by analyzing data sets that have been used in previous works dealing with model diagnostics, and a real data set pertaining to end-of-trading-day share values of the College Retirement and Equities Funds Growth and Stock accounts. Simulation results are presented indicating the sensitivity of the procedure in detecting model violations under a variety of situations.

ABSTRACT.For the usual linear model, bearing the plausibility of a redundant subset of parameters, pre-test and Stein-rule estimators based on the trimmed least squares estimation theory are considered. Compared to parallel M-estimators, proposed L-estimators are computationally simpler and are scale-equivariant too. In the light of asymptotic distributional risks, the relative (risk-)efficiency results for these trimmed L-estimators and their improved versions are studied in detail. Positive-rule L-estimators are also considered in this context. 1. INTRODUCTION. Consider

Minimum risk equivariant estimator in linear regression model

This document contains practicals to accompany the book Statistical Models (Davison, 2003), to which references below to Examples, Tables, and Figures refer. It is taken for granted that you have access to a current version of the statistical package

In a heteroscedastic linear model, if there is no replication, the usual approach is to model the variance. A common situation is that where the variance is modelled as a function of the mean response, or of a subset of the explanatory variables. Maximum likelihood estimation of the variance parameter in such models is very sensitive to extreme data points. In this dissertation alternative 'bounded influence' methods are developed which overcome this problem. The model for the variance which is considered is given by the re1ationship cr. = exp[h'(T.)8], where h is a known vector-valued function and 8 is 1- 1--the unknown variance parameter. Using this model, there is a parallel between the problem of developing efficient bounded influence estimators for 8 and that of developing efficient bounded influence estimators for the regression parameter in the homoscedastic regression case. Extension of several such methods of estimating ~ to include the problem of estimating 8 is discussed. The method proposed by Krasker (1980) is generalized to include estimation of 8 and shown to be optimal in a certain sense. The question of existence of a solution to the estimating equations is discussed. The methods of Krasker and Welsch (1982) case are extended to cover estimation of 8. in the homoscedastic regression A three-stage weighted regression estimate based on these techniques is proposed and the associated inf1uence calculations are presented. The class of bounded influence regression estimators proposed by Mallows (1975) is considered. A necessary condition for a strongly optimal weight function is derived and the resulting estimator of ~ is generalized to obtain an estimator of 8. Mallows estimators are shown to possess a stability of variance which is absent in Krasker-We1sch estimators. A three-stage Mallows regression estimate is proposed and its influence function is given. Computation of the three-stage estimates is discussed, and the performance of these methods is evaluated by using them in the analysis of a number of data sets.

We propose a new approach to conditional quantile function estimation that combines both parametric and nonparametric techniques. At each design point, a global, possibly incorrect, pilot parametric model is locally adjusted through a kernel smoothing fit. The resulting quantile regression estimator behaves like a parametric one when the latter is correct and converges to the nonparametric solution as the parametric start deviates from the true underlying model. We give a Bahadur-type representation of the proposed estimator from which consistency and asymptotic normality are derived under α-mixing assumption. We also propose a practical bandwidth selector based on the plug-in principle and discuss the numerical implementation of the new estimator. Finally, we investigate the performance of the proposed method via simulations and we illustrate the methodology on a data example. KEY WORDS: Bias reduction; Local polynomial smoothing; Model misspecification; Robustness; It is known from the literature that regression function estimators based on least squares are optimal and are equivalent to the maximum likelihood estimators when errors follow a normal